4 1. Jacobi operators

Furthermore,

£Q(I,M)

denotes the set of sequences with only finitely many

values being nonzero and -^_(Z,M) denotes the set of sequences in ^(Z, M) which

are

£p

near ±00, respectively (i.e., sequences whose restriction to ^(±N, M) belongs

to £

P

(±N,M). Here N denotes the set of positive integers). Note that, according

to our definition, we have

(1.3) £0(I, M) C

£P(I,

M) C JP(I, M) C e°°(I, M), p q,

with equality holding if and only if / is finite (assuming dimM 0).

In addition, if M is a (separable) Hilbert space with scalar product (.,..)M,

then the same is true for

£2(I,

M) with scalar product and norm defined by

n£l

(1-4) 11/1 1 = II/II2 = v ^ 7 . f,gef(I,M).

For what follows we will choose / = Z for simplicity. However, straightforward

modifications can be made to accommodate the general case / C Z.

During most of our investigations we will be concerned with difference expres-

sions, that is, endomorphisms of £(Z);

R: t(Z) -+ £(Z)

U'5)

f » Rf

(we reserve the name difference operator for difference expressions defined on a

subset of

£2(Z)).

Any difference expression R is uniquely determined by its corre-

sponding matrix representation

(1.6) ( # ( r a , n ) )

m n e Z

, R(m,ri) = (RSn)(m) = (8m,R8n),

where

(1.7) 6n{m) = 6min = i

1 n = m

is the canonical basis of £(Z). The order of R is the smallest nonnegative integer

N = N++N_ such that R{m, n) = 0 for all m, n with n — m N+ and m — n 7V_.

If no such number exists, the order is infinite.

We call R symmetric (resp. skew-symmetric) if R(m, n) = R(n, m) (resp.

R{m,n) = —R(n,m)).

Maybe the simplest examples for a difference expression are the shift expres-

sions

(1.8)

(S±f)(n)=f(n±l).

They are of particular importance due to the fact that their powers form a basis

for the space of all difference expressions (viewed as a module over the ring £{T)).

Indeed, we have

(1.9) i? = £ ( . , . +

fc)(S+)fe,

( 5

±

) "

1

= 5

T

.

kez

Here i?(.,. + k) denotes the multiplication expression with the sequence (R(n,n +

k))n£Zi that is, i?(.,. + k) : f(n) i-» R(n,n + k)f(n). In order to simplify notation

we agree to use the short cuts

(1.10)

f±=S±f, f++=S+S+f,

etc.,